Integrated circuits take a multitude forms, including digital memory chips, microprocessors, application specific integrated circuits (ASICs), application specific standard products (ASSPs), field-programmable gate arrays (FPGAs), hardware security modules (HSMs), and more. For many systems containing ICs, it is important to protect the electronically stored and/or processed data, including, but not limited to, computer access control, military weapons systems, medical information, vehicle control, secure communications, and payment transaction processing. The security for the data these systems process often relies on cryptographic operations based on secret keys stored in memory or other circuitry, which are then used to cryptographically secure sensitive data from unauthorized access.
Information leaked from cryptographic circuits can be analyzed to determine keys or other secret data being processed. The present invention provides methods and apparatuses to reduce information leakage during modular exponentiation and/or elliptic curve point multiplication.